An atomic clock works like a conventional clock but the time-base of the clock, instead of being an oscillating mass as in a pendulum clock, is based on the properties of atoms when transitioning between different energy states.
When an atom is excited by an external energy source, it goes to a higher energy state. Then, from this state, it goes to a lower energy state. In this transition, the atom releases energy at a very precise frequency which is characteristic of the type of atom.
All that is needed for building a good clock is a way of detecting this frequency that will be used as an input to a counter.
The electronics used by the space atomic clocks is one of the biggest difference from their ground clocks. They are composed of space-qualified electronics, that allows them to survive to the launch and to the space environmental conditions (thermal, radiation, etc.).
Galileo satellites carry Passive Hydrogen Massers (master clocks) and Rubidium Clocks that will be used as backup.